1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
|
(*************************************************************)
(* This file is distributed under the terms of the *)
(* GNU Lesser General Public License Version 2.1 *)
(*************************************************************)
(* Benjamin.Gregoire@inria.fr Laurent.Thery@inria.fr *)
(*************************************************************)
(***********************************************************************
Summation.v from Z to Z
*********************************************************************)
Require Import Coq.Arith.Arith.
Require Import Coq.setoid_ring.ArithRing.
Require Import Coqprime.ListAux.
Require Import Coq.ZArith.ZArith.
Require Import Coqprime.Iterator.
Require Import Coqprime.ZProgression.
Open Scope Z_scope.
(* Iterated Sum *)
Definition Zsum :=
fun n m f =>
if Zle_bool n m
then iter 0 f Zplus (progression Zsucc n (Zabs_nat ((1 + m) - n)))
else iter 0 f Zplus (progression Zpred n (Zabs_nat ((1 + n) - m))).
Hint Unfold Zsum .
Lemma Zsum_nn: forall n f, Zsum n n f = f n.
intros n f; unfold Zsum; rewrite Zle_bool_refl.
replace ((1 + n) - n) with 1; auto with zarith.
simpl; ring.
Qed.
Theorem permutation_rev: forall (A:Set) (l : list A), permutation (rev l) l.
intros a l; elim l; simpl; auto.
intros a1 l1 Hl1.
apply permutation_trans with (cons a1 (rev l1)); auto.
change (permutation (rev l1 ++ (a1 :: nil)) (app (cons a1 nil) (rev l1))); auto.
Qed.
Lemma Zsum_swap: forall (n m : Z) (f : Z -> Z), Zsum n m f = Zsum m n f.
intros n m f; unfold Zsum.
generalize (Zle_cases n m) (Zle_cases m n); case (Zle_bool n m);
case (Zle_bool m n); auto with arith.
intros; replace n with m; auto with zarith.
3:intros H1 H2; contradict H2; auto with zarith.
intros H1 H2; apply iter_permutation; auto with zarith.
apply permutation_trans
with (rev (progression Zsucc n (Zabs_nat ((1 + m) - n)))).
apply permutation_sym; apply permutation_rev.
rewrite Zprogression_opp; auto with zarith.
replace (n + Z_of_nat (pred (Zabs_nat ((1 + m) - n)))) with m; auto.
replace (Zabs_nat ((1 + m) - n)) with (S (Zabs_nat (m - n))); auto with zarith.
simpl.
rewrite inj_Zabs_nat; auto with zarith.
rewrite Zabs_eq; auto with zarith.
replace ((1 + m) - n) with (1 + (m - n)); auto with zarith.
cut (0 <= m - n); auto with zarith; unfold Zabs_nat.
case (m - n); auto with zarith.
intros p; case p; simpl; auto with zarith.
intros p1 Hp1; rewrite nat_of_P_xO; rewrite nat_of_P_xI;
rewrite nat_of_P_succ_morphism.
simpl; repeat rewrite plus_0_r.
repeat rewrite <- plus_n_Sm; simpl; auto.
intros p H3; contradict H3; auto with zarith.
intros H1 H2; apply iter_permutation; auto with zarith.
apply permutation_trans
with (rev (progression Zsucc m (Zabs_nat ((1 + n) - m)))).
rewrite Zprogression_opp; auto with zarith.
replace (m + Z_of_nat (pred (Zabs_nat ((1 + n) - m)))) with n; auto.
replace (Zabs_nat ((1 + n) - m)) with (S (Zabs_nat (n - m))); auto with zarith.
simpl.
rewrite inj_Zabs_nat; auto with zarith.
rewrite Zabs_eq; auto with zarith.
replace ((1 + n) - m) with (1 + (n - m)); auto with zarith.
cut (0 <= n - m); auto with zarith; unfold Zabs_nat.
case (n - m); auto with zarith.
intros p; case p; simpl; auto with zarith.
intros p1 Hp1; rewrite nat_of_P_xO; rewrite nat_of_P_xI;
rewrite nat_of_P_succ_morphism.
simpl; repeat rewrite plus_0_r.
repeat rewrite <- plus_n_Sm; simpl; auto.
intros p H3; contradict H3; auto with zarith.
apply permutation_rev.
Qed.
Lemma Zsum_split_up:
forall (n m p : Z) (f : Z -> Z),
( n <= m < p ) -> Zsum n p f = Zsum n m f + Zsum (m + 1) p f.
intros n m p f [H H0].
case (Zle_lt_or_eq _ _ H); clear H; intros H.
unfold Zsum; (repeat rewrite Zle_imp_le_bool); auto with zarith.
assert (H1: n < p).
apply Zlt_trans with ( 1 := H ); auto with zarith.
assert (H2: m < 1 + p).
apply Zlt_trans with ( 1 := H0 ); auto with zarith.
assert (H3: n < 1 + m).
apply Zlt_trans with ( 1 := H ); auto with zarith.
assert (H4: n < 1 + p).
apply Zlt_trans with ( 1 := H1 ); auto with zarith.
replace (Zabs_nat ((1 + p) - (m + 1)))
with (minus (Zabs_nat ((1 + p) - n)) (Zabs_nat ((1 + m) - n))).
apply iter_progression_app; auto with zarith.
apply inj_le_rev.
(repeat rewrite inj_Zabs_nat); auto with zarith.
(repeat rewrite Zabs_eq); auto with zarith.
rewrite next_n_Z; auto with zarith.
rewrite inj_Zabs_nat; auto with zarith.
rewrite Zabs_eq; auto with zarith.
apply inj_eq_rev; auto with zarith.
rewrite inj_minus1; auto with zarith.
(repeat rewrite inj_Zabs_nat); auto with zarith.
(repeat rewrite Zabs_eq); auto with zarith.
apply inj_le_rev.
(repeat rewrite inj_Zabs_nat); auto with zarith.
(repeat rewrite Zabs_eq); auto with zarith.
subst m.
rewrite Zsum_nn; auto with zarith.
unfold Zsum; generalize (Zle_cases n p); generalize (Zle_cases (n + 1) p);
case (Zle_bool n p); case (Zle_bool (n + 1) p); auto with zarith.
intros H1 H2.
replace (Zabs_nat ((1 + p) - n)) with (S (Zabs_nat (p - n))); auto with zarith.
replace (n + 1) with (Zsucc n); auto with zarith.
replace ((1 + p) - Zsucc n) with (p - n); auto with zarith.
apply inj_eq_rev; auto with zarith.
rewrite inj_S; (repeat rewrite inj_Zabs_nat); auto with zarith.
(repeat rewrite Zabs_eq); auto with zarith.
Qed.
Lemma Zsum_S_left:
forall (n m : Z) (f : Z -> Z), n < m -> Zsum n m f = f n + Zsum (n + 1) m f.
intros n m f H; rewrite (Zsum_split_up n n m f); auto with zarith.
rewrite Zsum_nn; auto with zarith.
Qed.
Lemma Zsum_S_right:
forall (n m : Z) (f : Z -> Z),
n <= m -> Zsum n (m + 1) f = Zsum n m f + f (m + 1).
intros n m f H; rewrite (Zsum_split_up n m (m + 1) f); auto with zarith.
rewrite Zsum_nn; auto with zarith.
Qed.
Lemma Zsum_split_down:
forall (n m p : Z) (f : Z -> Z),
( p < m <= n ) -> Zsum n p f = Zsum n m f + Zsum (m - 1) p f.
intros n m p f [H H0].
case (Zle_lt_or_eq p (m - 1)); auto with zarith; intros H1.
pattern m at 1; replace m with ((m - 1) + 1); auto with zarith.
repeat rewrite (Zsum_swap n).
rewrite (Zsum_swap (m - 1)).
rewrite Zplus_comm.
apply Zsum_split_up; auto with zarith.
subst p.
repeat rewrite (Zsum_swap n).
rewrite Zsum_nn.
unfold Zsum; (repeat rewrite Zle_imp_le_bool); auto with zarith.
replace (Zabs_nat ((1 + n) - (m - 1))) with (S (Zabs_nat (n - (m - 1)))).
rewrite Zplus_comm.
replace (Zabs_nat ((1 + n) - m)) with (Zabs_nat (n - (m - 1))); auto with zarith.
pattern m at 4; replace m with (Zsucc (m - 1)); auto with zarith.
apply f_equal with ( f := Zabs_nat ); auto with zarith.
apply inj_eq_rev; auto with zarith.
rewrite inj_S.
(repeat rewrite inj_Zabs_nat); auto with zarith.
(repeat rewrite Zabs_eq); auto with zarith.
Qed.
Lemma Zsum_ext:
forall (n m : Z) (f g : Z -> Z),
n <= m ->
(forall (x : Z), ( n <= x <= m ) -> f x = g x) -> Zsum n m f = Zsum n m g.
intros n m f g HH H.
unfold Zsum; auto.
unfold Zsum; (repeat rewrite Zle_imp_le_bool); auto with zarith.
apply iter_ext; auto with zarith.
intros a H1; apply H; auto; split.
apply Zprogression_le_init with ( 1 := H1 ).
cut (a < Zsucc m); auto with zarith.
replace (Zsucc m) with (n + Z_of_nat (Zabs_nat ((1 + m) - n))); auto with zarith.
apply Zprogression_le_end; auto with zarith.
rewrite inj_Zabs_nat; auto with zarith.
(repeat rewrite Zabs_eq); auto with zarith.
Qed.
Lemma Zsum_add:
forall (n m : Z) (f g : Z -> Z),
Zsum n m f + Zsum n m g = Zsum n m (fun (i : Z) => f i + g i).
intros n m f g; unfold Zsum; case (Zle_bool n m); apply iter_comp;
auto with zarith.
Qed.
Lemma Zsum_times:
forall n m x f, x * Zsum n m f = Zsum n m (fun i=> x * f i).
intros n m x f.
unfold Zsum. case (Zle_bool n m); intros; apply iter_comp_const with (k := (fun y : Z => x * y)); auto with zarith.
Qed.
Lemma inv_Zsum:
forall (P : Z -> Prop) (n m : Z) (f : Z -> Z),
n <= m ->
P 0 ->
(forall (a b : Z), P a -> P b -> P (a + b)) ->
(forall (x : Z), ( n <= x <= m ) -> P (f x)) -> P (Zsum n m f).
intros P n m f HH H H0 H1.
unfold Zsum; rewrite Zle_imp_le_bool; auto with zarith; apply iter_inv; auto.
intros x H3; apply H1; auto; split.
apply Zprogression_le_init with ( 1 := H3 ).
cut (x < Zsucc m); auto with zarith.
replace (Zsucc m) with (n + Z_of_nat (Zabs_nat ((1 + m) - n))); auto with zarith.
apply Zprogression_le_end; auto with zarith.
rewrite inj_Zabs_nat; auto with zarith.
(repeat rewrite Zabs_eq); auto with zarith.
Qed.
Lemma Zsum_pred:
forall (n m : Z) (f : Z -> Z),
Zsum n m f = Zsum (n + 1) (m + 1) (fun (i : Z) => f (Zpred i)).
intros n m f.
unfold Zsum.
generalize (Zle_cases n m); generalize (Zle_cases (n + 1) (m + 1));
case (Zle_bool n m); case (Zle_bool (n + 1) (m + 1)); auto with zarith.
replace ((1 + (m + 1)) - (n + 1)) with ((1 + m) - n); auto with zarith.
intros H1 H2; cut (exists c , c = Zabs_nat ((1 + m) - n) ).
intros [c H3]; rewrite <- H3.
generalize n; elim c; auto with zarith; clear H1 H2 H3 c n.
intros c H n; simpl; eq_tac; auto with zarith.
eq_tac; unfold Zpred; auto with zarith.
replace (Zsucc (n + 1)) with (Zsucc n + 1); auto with zarith.
exists (Zabs_nat ((1 + m) - n)); auto.
replace ((1 + (n + 1)) - (m + 1)) with ((1 + n) - m); auto with zarith.
intros H1 H2; cut (exists c , c = Zabs_nat ((1 + n) - m) ).
intros [c H3]; rewrite <- H3.
generalize n; elim c; auto with zarith; clear H1 H2 H3 c n.
intros c H n; simpl; (eq_tac; auto with zarith).
eq_tac; unfold Zpred; auto with zarith.
replace (Zpred (n + 1)) with (Zpred n + 1); auto with zarith.
unfold Zpred; auto with zarith.
exists (Zabs_nat ((1 + n) - m)); auto.
Qed.
Theorem Zsum_c:
forall (c p q : Z), p <= q -> Zsum p q (fun x => c) = ((1 + q) - p) * c.
intros c p q Hq; unfold Zsum.
rewrite Zle_imp_le_bool; auto with zarith.
pattern ((1 + q) - p) at 2.
rewrite <- Zabs_eq; auto with zarith.
rewrite <- inj_Zabs_nat; auto with zarith.
cut (exists r , r = Zabs_nat ((1 + q) - p) );
[intros [r H1]; rewrite <- H1 | exists (Zabs_nat ((1 + q) - p))]; auto.
generalize p; elim r; auto with zarith.
intros n H p0; replace (Z_of_nat (S n)) with (Z_of_nat n + 1); auto with zarith.
simpl; rewrite H; ring.
rewrite inj_S; auto with zarith.
Qed.
Theorem Zsum_Zsum_f:
forall (i j k l : Z) (f : Z -> Z -> Z),
i <= j ->
k < l ->
Zsum i j (fun x => Zsum k (l + 1) (fun y => f x y)) =
Zsum i j (fun x => Zsum k l (fun y => f x y) + f x (l + 1)).
intros; apply Zsum_ext; intros; auto with zarith.
rewrite Zsum_S_right; auto with zarith.
Qed.
Theorem Zsum_com:
forall (i j k l : Z) (f : Z -> Z -> Z),
Zsum i j (fun x => Zsum k l (fun y => f x y)) =
Zsum k l (fun y => Zsum i j (fun x => f x y)).
intros; unfold Zsum; case (Zle_bool i j); case (Zle_bool k l); apply iter_com;
auto with zarith.
Qed.
Theorem Zsum_le:
forall (n m : Z) (f g : Z -> Z),
n <= m ->
(forall (x : Z), ( n <= x <= m ) -> (f x <= g x )) ->
(Zsum n m f <= Zsum n m g ).
intros n m f g Hl H.
unfold Zsum; rewrite Zle_imp_le_bool; auto with zarith.
unfold Zsum;
cut
(forall x,
In x (progression Zsucc n (Zabs_nat ((1 + m) - n))) -> ( f x <= g x )).
elim (progression Zsucc n (Zabs_nat ((1 + m) - n))); simpl; auto with zarith.
intros x H1; apply H; split.
apply Zprogression_le_init with ( 1 := H1 ); auto.
cut (x < m + 1); auto with zarith.
replace (m + 1) with (n + Z_of_nat (Zabs_nat ((1 + m) - n))); auto with zarith.
apply Zprogression_le_end; auto with zarith.
rewrite inj_Zabs_nat; auto with zarith.
rewrite Zabs_eq; auto with zarith.
Qed.
Theorem iter_le:
forall (f g: Z -> Z) l, (forall a, In a l -> f a <= g a) ->
iter 0 f Zplus l <= iter 0 g Zplus l.
intros f g l; elim l; simpl; auto with zarith.
Qed.
Theorem Zsum_lt:
forall n m f g,
(forall x, n <= x -> x <= m -> f x <= g x) ->
(exists x, n <= x /\ x <= m /\ f x < g x) ->
Zsum n m f < Zsum n m g.
intros n m f g H (d, (Hd1, (Hd2, Hd3))); unfold Zsum; rewrite Zle_imp_le_bool; auto with zarith.
cut (In d (progression Zsucc n (Zabs_nat (1 + m - n)))).
cut (forall x, In x (progression Zsucc n (Zabs_nat (1 + m - n)))-> f x <= g x).
elim (progression Zsucc n (Zabs_nat (1 + m - n))); simpl; auto with zarith.
intros a l Rec H0 [H1 | H1]; subst; auto.
apply Zle_lt_trans with (f d + iter 0 g Zplus l); auto with zarith.
apply Zplus_le_compat_l.
apply iter_le; auto.
apply Zlt_le_trans with (f a + iter 0 g Zplus l); auto with zarith.
intros x H1; apply H.
apply Zprogression_le_init with ( 1 := H1 ); auto.
cut (x < m + 1); auto with zarith.
replace (m + 1) with (n + Z_of_nat (Zabs_nat ((1 + m) - n))); auto with zarith.
apply Zprogression_le_end with ( 1 := H1 ); auto with arith.
rewrite inj_Zabs_nat; auto with zarith.
rewrite Zabs_eq; auto with zarith.
apply in_Zprogression.
rewrite inj_Zabs_nat; auto with zarith.
rewrite Zabs_eq; auto with zarith.
Qed.
Theorem Zsum_minus:
forall n m f g, Zsum n m f - Zsum n m g = Zsum n m (fun x => f x - g x).
intros n m f g; apply trans_equal with (Zsum n m f + (-1) * Zsum n m g); auto with zarith.
rewrite Zsum_times; rewrite Zsum_add; auto with zarith.
Qed.
|
